A coupling need not be deterministic.
For instance, we can have \(Z= T(\Xi , X)\sim \pi _{X\vert y^*}\)  for some deterministic transformation \(T\)  and additional random variable \(\Xi \) .
As a function of \(X\)  alone,
we can think of \(T\) 
as a “stochastic” map.
In this
work,
we consider
non-deterministic couplings induced by a
map
\(T:\mathbb {R}^d \times \mathbb {R}^n \rightarrow \mathbb {R}^n\)  such that \(Z= T(Y, X)\) , where
\(Y\)  represents the data random variable of the Bayesian inference problem.
These
couplings can be used for
posterior sampling like their deterministic
counterpart:
if \((y^1,x^1),\ldots ,(y^M,x^M)\)  are i.i.d. samples from the joint distribution \(\pi _{Y,X}\) , then
\(T(y^1,x^1),\ldots , T(y^M,x^M)\)  are also
i.i.d. samples from the posterior.
In fact, the transformation \(T\)  is
a transport map
that pushes forward \(\pi _{Y,X}\)  to
the
posterior.
Inference with non-deterministic couplings is interesting because we can build estimators for
\(T\)  using only
convex optimization.
We will see in Section
 that non-deterministic couplings
are linked to nonlinear generalizations of the stochastic EnKF [1]}.
